# The Spectre of Math

## May 11, 2010

### Yay! End of semester!

Filed under: Mathematics,Personal,Teaching — jlebl @ 9:01 pm

It is finally end of semester. Still have to grade some finals, but other than that, I’m done. It’s also the end of my time at UIUC, next year I’m off to UCSD for a year.

I finished off, at least in a rough draft sort of way, my notes on Hermitian forms and CR geometry, a half semester minicourse I gave at UIUC. Perhaps those will be useful to others. I can’t really say that they will be “bug free,” I am sure there are many typos. There are even some previously unpublished results (though probably previously known). For example, suppose that you have a real analytic function $r(z,\bar{z})$ such that the series (in multiindex notation)

$\sum_{\alpha,\beta} c_{\alpha \beta} z^\alpha \bar{z}^\beta$

converges in the neighbourhood of the unit polydisc (converges absolutely at the point $z = (1,1,1,\ldots,1)$). Then the Hermitian matrix $C = [ c_{\alpha\beta} ]_{\alpha\beta}$ defines a bounded operator on $\ell^2$ (actually it defines a compact operator). For a long time I was always doing all these matrix operations with the coefficient matrices formally. And then you have to be very careful. Once you rescale so that the coefficient matrix is a compact operator, then it’s just like doing linear algebra, and all the subtleties go away.

I also kept updating the differential equations notes like crazy this semester. The students could have gotten up to 3% extra credit for finding errors, but only a handful did. Majority of typos I found myself. The good news is that the notes, especially chapters 0,1,2, and 4 are reasonably bug free. Chapters 3,5,6 also got some bug-spray treatment, but not as much. I still have a bunch of minor grammatical errors pending and I’ll wait for a week maybe before putting out another version just in case somebody still finds some more.